3.226 \(\int \frac {1}{\sqrt {d x} (a+b \cos ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{\sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),x)

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2),x]

[Out]

Defer[Int][1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx &=\int \frac {1}{\sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 10.43, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2),x]

[Out]

Integrate[1/(Sqrt[d*x]*(a + b*ArcCos[c*x])^2), x]

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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x}}{b^{2} d x \arccos \left (c x\right )^{2} + 2 \, a b d x \arccos \left (c x\right ) + a^{2} d x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x)/(b^2*d*x*arccos(c*x)^2 + 2*a*b*d*x*arccos(c*x) + a^2*d*x), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(d*x)*(b*arccos(c*x) + a)^2), x)

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \arccos \left (c x \right )\right )^{2} \sqrt {d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),x)

[Out]

int(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\frac {1}{2} \, {\left (b^{2} c d x \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a b c d x\right )} \sqrt {d} \int \frac {{\left (c^{2} x^{2} + 1\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {x}}{a b c^{3} d x^{4} - a b c d x^{2} + {\left (b^{2} c^{3} d x^{4} - b^{2} c d x^{2}\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )}\,{d x} - \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {d} \sqrt {x}}{b^{2} c d x \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a b c d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),x, algorithm="maxima")

[Out]

-((b^2*c*d*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*d*x)*sqrt(d)*integrate(1/2*(c^2*x^2 + 1)*sqrt(
c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(a*b*c^3*d*x^4 - a*b*c*d*x^2 + (b^2*c^3*d*x^4 - b^2*c*d*x^2)*arctan2(sqrt(c*x
+ 1)*sqrt(-c*x + 1), c*x)), x) - sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*sqrt(x))/(b^2*c*d*x*arctan2(sqrt(c*x + 1
)*sqrt(-c*x + 1), c*x) + a*b*c*d*x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,\sqrt {d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*acos(c*x))^2*(d*x)^(1/2)),x)

[Out]

int(1/((a + b*acos(c*x))^2*(d*x)^(1/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*acos(c*x))**2/(d*x)**(1/2),x)

[Out]

Integral(1/(sqrt(d*x)*(a + b*acos(c*x))**2), x)

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